USAGE: Words for fungi (mushrooms/toadstools/champignons/etc)

We seem to be agreed that there is a general, broad category called "fungi"
(singular "fungus"), which includes both umbrella-shaped and
non-umbrella-shaped, edible and inedible varieties.
We are further in agreement that edible umbrella-shaped fungi are
"mushrooms".
Past this point, disagreement sets in. Inedible umbrella-shaped fungi
are most commonly considered "toadstools", but some use either
"mushroom" or "toadstool" to refer to all umbrella-shaped fungi
regardless of edibility. (Hm. Does anyone use the terms completely
interchangeably?)
Is there any general term more specific than "fungus/i" to
refer specifically to the non-umbrella-shaped varieties, or to
edible/inedible varieties regardless of shape?
-Marcos